An Algebraic Reformulation of the Koide Relation via the Cube Roots of Unity

Abstract The Koide formula Q = (mₑ + mμ + mτ) / (√mₑ + √mμ + √mτ)² takes the value 2/3 to five decimal places when current Particle Data Group values for the charged-lepton masses are substituted (Qₒᵇˢ = 0.666664, consistent with 2/3 within the τ-mass uncertainty). This paper does not propose a derivation of that value from a dynamical model; it presents an algebraic reformulation, together with one clearly labelled hypothesis. The charged-lepton mass-square-root amplitudes are written as a phase-symmetric triple along the three cube roots of unity (1, ω, ω²), ω = exp(2πi/3). Under this representation Q is shown to be algebraically identical to the ratio of complex-conjugate-pair weight to total weight in the triple, equal to 2/3, and equivalently to the phase-weighted-sum identity |L|² = (3/4)Σm. In the one-parameter (η, δ) form Q = (1 + 2η²)/3 the Koide condition is η² = 1/2, independent of scale and of δ, so that δ is the single residual degree of freedom that sets the spectrum; the charged leptons select δ = 2/9 (residual 2.5×10⁻⁶ under the explicit fitting convention of Section 4.5, with 2/9 inside the 1σ interval), reproducing e, μ, τ to better than 0.003%. We quantify, under explicit priors, how sharply the spectrum selects this rational value, and we add one hypothesis-level extension: if a single carrier law |L|² = (3/4)Σm holds on symmetric configurations, positive semidefiniteness of the Gram matrix forces Q ≤ 2/3, with equality exactly at the cube-root alignment; the measured precision of Q then bounds the charged-lepton internal misalignment to φ ≤ 0.19° (1σ), and any family with Q > 2/3 would refute the hypothesis. The reformulation itself does not predict Q or δ: the masses are an input. We record, for the avoidance of an overstatement of novelty, that the value δ = 2/9 and the underlying Z₃ (cosine) parametrization are already present in the literature (Brannen 2006; Żenczykowski 2012, 2013, which use 1 and 2/9 explicitly); priority for the value belongs there. What remains unclaimed by the entire field is a derivation of 2/9 from first principles. We compare the reformulation with existing approaches and discuss the open status of the quark and neutrino sectors. A post-hoc coordinate reading of the measured non-lepton triples is recorded (Section 5.5) — the pion triplet at an exactly geometric phase, both quark families excluded from symmetric configurations by the bound — together with the explicit statement that the parametrization is universal by construction, so that its empirical content lies entirely in the bound, the recurring special values, and the clustering. Keywords: Koide formula; charged-lepton masses; cube roots of unity; phase-symmetric triples; Z₃ parametrization; algebraic reformulation; alignment bound.